Present Value

Present value is the current value of a future sum

If you receive $100 one year from now, what is the present value (PV) of that $100 if your opportunity cost is 6%?

Int = 6

N = 1 FV = 100

PV = 94.34

00 1 1

PV = PV = FV = 100 FV = 100

If you receive $100 five years from now, what is the present value (PV) of that $100 if your opportunity cost is 6%?

PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $ 74.73

00 5 5

PV = PV = FV = 100 FV = 100

Example

What is the present value (PV) of $500 to be received 10 years from now if the opportunity cost is 6%?

PV = FV / (1 + i)n

PV = 500 / (1.06)10

= 500(0.558) = $279

00 10 10

PV = PV = FV = 500 FV = 500

Periods per year = 12 FV = 500 I = 9.6 PV = -100 N = ? months

Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500?

00 ? ?

PV = -100PV = -100 FV = 500 FV = 500

Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500?

PV = FV / (1 + i)n

100 = 500 / (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)1.60944 = .007968 N N = 202 months

The present value (PV) of $500 to be received 10 years from now is $279

Investment that has two cash flows in different time periods

What is the present value (PV) of

an investment that results in $500 to be received in 5 years and $1000 to be received in 10 years if the opportunity cost is 4%?

Present value (PV) is the sum of the following:

500 / (1.04)5

and 1000 / (1.04)10

= 500(0.822) + 1000(0.676)= 411 + 676= $1087

Annuities

Annuities

Annuity

A sequence of equal cash flows, occurring at the end of each period

Annuity

A sequence of equal cash flows, occurring at the end of each period

0 1 2 3 4

Annuities

Examples of Annuities:

If you buy a bond with interest paid out semi-annually, you will receive equal semi-annual coupon interest payments over the life of the bond

If you borrow money to buy a house or a car, you will pay a stream of equal payments

Ordinary Annuity

Cash flows of an ordinary annuity occur at the end of each period

Annuity Due

Cash flows of an annuity due occur at the beginning of each period

Compound Annuity

Compound annuity involves depositing or investing an equal sum of money at the end of each year for a certain number of years

If we invest $1,000 at the end of each year at 8% for purchasing a car. How much would you have after 3 years ?

Periods / year = 1 I = 8 N = 3

PMT = -1,000

FV = $3246.40

If you invest $1,000 each year at 8%, how much would you have after 3 years?

0 1 2 3

10001000 1000 1000 1000 1000

Time line

A horizontal line on which the present time period (t=0) is at the left most end

Future time periods are shown along the line moving from left to right

Time line

The amounts of cash flow are shown below the line

Positive values represent cash inflows and negative values represent cash outflows at various time periods

Future value of compound annuity

The future value of a compound annuity is as follows:

1

0

(1 )n

tn

t

FV PMT i

The future-value of the annuity at the end of the year n is denoted by

The amount of annuity payment deposited at the end of each year is denoted by PMT

i is the annual rate of interest

n is the number of years for which the annuity will last

nFV

Future value of compound annuity

The future value of a compound annuity can be expressed as follows:

,( )n i nFV PMT FVIFA

Example

We need an amount of $10,000 for university education in 8 years. How much do we need to deposit at the end of each year in the bank at 6% interest to have the money ready for payment to the University?

8 1

0

10000 (1 0.06)t

t

PMT

6%,810000 ( )yrsPMT FVIFA

10000 (9.897)

$1010.41

PMT

PMT

Present value of an annuity

The present value of an annuity is as follows:

1

1

(1 )

n

tt

PV PMTi

The present value of the annuity

The amount of annuity received at the end of each year is denoted by PMT

i is the discount rate

n is the number of years for which the annuity will last

Present value of an annuity

The present value of an annuity can be expressed as follows:

,( )i nPV PMT PVIFA

Example

What is the present value of a 10 year $1000 annuity discounted back to the present at 5 percent?

10

1

11000

(1 0.05)tt

PV

1000(7.722)

$7,722

PV

Annuity Due

Annuity Due

Cash flows of an annuity due occur at the beginning of each period

Future value of an annuity due

As an annuity due, shifts the payments from the end of the year to the beginning of the year, we compound the cash flows for an another year

Future value of an annuity due

Future value of an annuity due is computed as follows:

,( . .) ( )(1 )n i nFV A D PMT FVIFA i

Periods / year = 1 I = 8 N = 3

PMT = -1,000

FV = 3246.40(1.08)

If you invest $1,000 each year in an annuity due at 8%, how much would you have after 3 years?

0 1 2 3

10001000 1000 1000 1000 1000

Future value of an annuity due

Future value of this annuity due is $3,506.11

In an annuity due, compounding is for 1 additional year

Present value of an annuity due

Present value of an ordinary annuity is multiplied by (1+i), which cancels out 1 year’s discounting:

,( )(1 )i nPV PMT PVIFA i

Example

What is the present value of a 10 year $1000 annuity due discounted back to the present at 5 percent?

1000(7.722)(1.05)

$8108.1

PV

In the discounting of an annuity due, the cash flows are discounted for 1 less year

Amortized Loan

Amortized Loan

An amortized loan is a loan that is repaid in equal installments over time

ExampleWe borrow US$ 20,000 for renovation of a house. The loan has to be repaid in 4 equal installments at the end of each of the 4 years. The lender charges an interest rate of 15% on the outstanding loan. What is the installment associated with repayment of the loan?

The installment for repaying the principal and interest on the loan in 4 years is US$7005.25

4

1

120000

(1 0.15)tt

PMT

20000 (2.855)

7005.25

PMT

PMT

Uneven cash flows

A project involving uneven cash flows over several years

For example, investments in fixed assets

We discount each of the cash flows back to the present.

We add the positive cash flows and subtract the negative cash flows

Perpetuity

A perpetuity is an investment that pays a constant permanent currency amount every year

The equation representing a perpetuity is as follows:

PPPV

i

Example

We have invested in a US$ 1000 perpetuity. Our discount rate is 9 percent. What is the present value of the perpetuity?

The present value of the perpetuity is as follows:

1000

0.09$11,111.11

PV

US